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Originally Posted by Yakk
Actually, I want to describe the integers and plus and times.
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If you are talking about arithmetic (I'm not sure what you are doing there by including the two operations, unless you mean the ring of integers), then I can only point to <a href="http://mathworld.wolfram.com/GoedelsIncompletenessTheorem.html">MathWorld</a>, which contains the following quotation.
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Gerhard Gentzen showed that the consistency and completeness of arithmetic can be proved if transfinite induction is used. However, this approach does not allow proof of the consistency of all mathematics.
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I am not talking about arithmetic though---Peano or otherwise, nor set theory. It is enough, in my opinion, to assume naive set theory to wave my hands and pretend in order to create my particular set.
However, you know more about Godel's work than I do I think, and it seems you are seeing something I am not, so I will defer the point. You are right that my approach is fast and loose.
Cheers.